Crack detection of the cantilever beam using new triple hybrid algorithms based on Particle Swarm Optimization

Abstract

The presence of cracks in a concrete structure reduces its performance and increases in the size of cracks result in the failure of the structure. Therefore, the accurate determination of crack characteristics, such as location and depth, is one of the key engineering issues for assessment of the reliability of structures. This paper deals with the inverse analysis of the crack detection problems using triple hybrid algorithms based on Particle Swarm Optimization (PSO); these hybrids are Particle Swarm Optimization-Genetic Algorithm-Firefly Algorithm (PSO-GA-FA), Particle Swarm Optimization-Grey Wolf Optimization-Firefly Algorithm (PSO-GWO-FA), and Particle Swarm Optimization-Genetic Algorithm-Grey Wolf Optimization (PSO-GA-GWO). A strong correlation exists between the changes in the natural frequency of a concrete beam and the crack parameters. Thus, the location and depth of a crack in a beam can be predicted by measuring its natural frequency. Hence, the measured natural frequency can be used as the input parameter of the algorithm. In this paper, this is applied to identify crack location and depth in a cantilever beam using the new hybrid algorithms. The results show that among the proposed triple hybrid algorithms, the PSO-GA-FA and PSO-GWO-FA algorithms are much more effective than PSO-GA-GWO algorithm for the crack detection.

Graphical abstract

Keywords

crack / cantilever beam / triple hybrid algorithms / Particle Swarm Optimization

1 Introduction

The existing crack within a structural component changes its local stiffness. So, vibration amplitudes and natural frequencies change due to the existence of such cracks. Therefore, it is feasible to determine crack location and depth by analysis of these changes. One of the methods of determining the crack location and depth is to utilize the mode shapes and the natural frequencies. An inverse method can be used for identification of structural damages. For this purpose, the optimization method is repeated to minimize or maximize the objective function so as to locate a crack. In this repeated operation, the unknown crack location can be located based on some parameters that can be updated using the optimization method to reach the best answer. Dimarogonas and Papadopoulos [1] and Qian et al. [2] prepared the stiffness matrix of the cracked beam and used intensity factors for investigation of the dynamic response characteristics such as the mode shapes and the natural frequencies. Nahvi and Jabbari [3] evaluated the failure intensity of cantilever cracked beams using the finite element method and experimental data. The lateral vibration of cracked Euler-Bernoulli beams was evaluated by Chondros et al. [4]. Also, Kim and Stubbs [5] investigated a feasible technique to calculate depths of cracks by utilizing modification in the natural frequencies. Orhan [6] studied the forced and free vibration analysis for recognizing cracks in a cantilever beam. Saavedra and Cuitino [7] explained the behavior of crack beams using the strain energy density function. Using a finite element manner, Zheng and Kessissoglou [8] investigated the natural frequencies and the mode shapes of a cracked beam. Rizos et al. [9] considered the flexural vibrations of a cantilever beam with surface crack. Sahoo and Maity [10] offered the hybrid Neuro-Genetic Algorithm (GA) to study crack parameters. Vakil-Baghmisheh et al. [11,12] investigated location and depth of cracks by utilizing the GA algorithm and hybrid techniques that compounded Nelder-Mead (NM) and Particle Swarm Optimization (PSO). Patil and Maiti [13] proposed the Transfer Matrix Technique for measurement of locations of multiplex cracks using the measurement of natural frequencies. Rosales et al. [14] developed a technique to solve the inverse problem of crack location by compounding neural network methods with power series. Moezi et al. [15] precisely evaluated the crack location and depth in the beam using the Modified Cuckoo Optimization Algorithm. Nandwana and Maiti [16] explained crack discovery in a stepped beam with cracks, using models such as torsional spring. Lele and Maiti [17] developed a technique for short beams by utilizing measured frequencies to recognize the crack. Viola et al. [18] studied the cracked beam by obtaining the stable mass and stiffness matrices. Rezanezhad et al. [19] used the extended finite element method (XFEM), and modeled crack growth in a natural porous environment (like Fontainebleau sandstone).

Investigation of crack-location of 3D and 2D piezoelectric structures by utilizing XFEM methodology in the inverse and the direct problem are presented in Ref. [20,22–25]. Rabczuk and Belytschko [26] presented a novel technique for treating crack propagation by particle methods. Also, a geometrically non-linear three-dimensional cohesive crack was expressed using the extended element-free Galerkin method for reinforced concrete structures [27]. Ghasemi et al. [28,29] investigated the behavior of flexoelectric composites using the computational design method, location Isogeometric Analysis (IGA), and point-wise density mapping techniques. Also, Ghasemi et al. [30,31] studied probabilistic multi-constraints optimization of cooling channels in ceramic matrix composites (CMC) and a reinforcement distribution optimizer based on Non-uniform Rational B-spline (NURBS) as a methodology for decreasing interfacial stresses in sandwich beams. Talebi et al. [32] presented multiscale modeling of three-dimensional crack and dislocation propagation. Zhou et al. [33–35] presented a Phase-Field Model (PFM) for simulating complex crack patterns including crack propagation, branching, and coalescence in rock and poroelastic media using an implicit time integration scheme and the Newton–Raphson iteration in commercial finite element software COMSOL. In previous research, the applied techniques were extended for crack recognition in beams. Therefore, various methods and algorithms have been used in recent years to identify the presence of cracks in beam structures. Tab.1 presents a short review of some researches and their applied techniques. In this paper, a procedure is applied to evaluate the locations and depths of cracks in cantilever beams, which improves the precision of crack detection. In this study, new hybrid algorithms are utilized for crack detection of cantilever beams. These triple hybrid algorithms combine GA, PSO, Grey Wolf Optimization (GWO), and Firefly Algorithm (FA). These algorithms are used to minimize the cost function to detect the locations and depths of cracks in cantilever beams. In other words, this paper uses Particle Swarm Optimization-Genetic Algorithm-Grey Wolf Optimization (PSO-GA-GWO), Particle Swarm Optimization-Genetic Algorithm-Firefly Algorithm (PSO-GA-FA), and Particle Swarm Optimization-Grey Wolf Optimization-Firefly Algorithm (PSO-GWO-FA) algorithms for evaluating tasks to obtain the best results. Finally, the results obtained from these triple algorithms are compared with other algorithms such as GA, GWO, FA, PSO, and Modified Particle Swarm Optimization (MPSO).

2 Modelling of the cracked beam

In this paper, the cantilever beam is considered as shown in Fig.1. This beam has length “L”, width “b”, depth of the crack “a” at changeable location L₁, and thickness “h”. The flexibility matrix is represented via stress intensity factors, and the existence of a transverse surface crack affects the dynamic efficiency of the construction. The relation between the stress intensity factors and the strain energy release rates at the crack segment has been presented by Jena and Parhi [36]. Also, by considering the reverse of the flexibility matrix [1], the local stiffness matrix can be obtained. The differential equations of the free vibration of an Euler-Bernoulli beam can be determined as:

(1)EI\frac{{\mathrm{\partial }}^{4}W}{\mathrm{\partial }{X}^4}+m\frac{{\mathrm{\partial }}^{2}W}{\mathrm{\partial }{t}^2}=0,

(2)\left(\frac{E}{\rho }\right)\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{X}^2}=\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{t}^2}.

Equation (1) applies to transverse vibration, and Eq. (2) applies to the longitudinal vibration; m is the unit mass of the length of the beam; W and V demonstrate the transverse and longitudinal movements; ρ is the density. The responses of the transverse vibration and the longitudinal vibration are determined according to the method applied in Ref. [37]. The cracked beam is separated into two segments of left and right (x \in [0,L₁]) and x \in (L₁,L]). Therefore, we have:

\begin{array}{rl}{W}_{\mathrm{R}}\left(x\right)=& C_{\mathrm{R}1}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\frac{\lambda x}{L}\right)+C_{\mathrm{R}2}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\frac{\lambda x}{L}\right)\\ & +C_{\mathrm{R}3}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\lambda x}{L}\right)+C_{\mathrm{R}4}\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\lambda x}{L}\right),\end{array}

\begin{array}{rl}{W}_{\mathrm{L}}\left(x\right)=& C_{\mathrm{L}1}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\frac{\lambda x}{L}\right)+C_{\mathrm{L}2}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\frac{\lambda x}{L}\right)\\ & +C_{\mathrm{L}3}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\lambda x}{L}\right)+C_{\mathrm{L}4}\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\lambda x}{L}\right),\end{array}

\begin{array}{}\text{(4a)}& V_{\mathrm{R}}\left(x\right)=C_{\mathrm{R}5}\mathrm{c}\mathrm{o}\mathrm{s}\left(\lambda \frac{x}{L}\right)+C_{\mathrm{R}6}\mathrm{s}\mathrm{i}\mathrm{n}\left(\lambda \frac{x}{L}\right),\end{array}

\begin{array}{}\text{(4b)}& V_{\mathrm{L}}\left(x\right)=C_{\mathrm{L}5}\mathrm{c}\mathrm{o}\mathrm{s}\left(\lambda \frac{x}{L}\right)+C_{\mathrm{L}6}\mathrm{s}\mathrm{i}\mathrm{n}\left(\lambda \frac{x}{L}\right).\end{array}

The term \lambda (in Eqs. (3a), (3b), (4a), and (4b)) is dependent on the natural frequency of the beam, (i.e., {\lambda }^4 = \rho A{\omega }^2/EI). Also, C_R,i = 1:6, and C_L,i = 1:6 are the unknown coefficients that can be defined using suitable boundary conditions and the continuity conditions at the cracked section. The relations can be expressed according to Ref. [37]. The equation of the system can be shown as |B| = 0 that contains the local stiffness matrix [36]. This is a function of the non-dimensional crack depth, natural frequency (ω), and function of the relative locations of the crack (α). Matrix B is shown explicitly in Appendix.

3 Objective function formulation for locating a crack by utilizing optimization algorithms

As mentioned in the above sections, recognizing variations of the natural frequencies of the cracked beams for a special location and depth of crack is an easy process. Evaluating the unknown location and depth of cracks repeatedly by utilizing an optimization algorithm is the purpose of the inverse technique, which leads to calculation of the real and the evaluated natural frequencies. The minimized objective function of the inverse problem can be considered as:

(5)\mathrm{M}\mathrm{i}\mathrm{n}f\left(l_{\mathrm{c}},d_{\mathrm{c}}\right)={\sum }_{i=1}^m\left[{\phi }_i\left(f_i^{\mathrm{d}}-f_i^{\mathrm{s}}\right)\right].

The optimization algorithm searches the location and depth of cracks in a manner in which the summation of variations among the evaluated and measured frequencies is minimized to zero. According to the restriction, it is assumed 0 < a < h and 0 < L1 < L. In Eq. (5), “m” is the number of natural frequencies, “\phi_i” is the ith weighting factor, “f_i^{\mathrm{d}}” is the ith demanded natural frequency of a cracked beam, “f_i^{\mathrm{s}}” refers to the ith natural frequency that is evaluated by the algorithm and utilized to estimate the objective function. In this paper, the first three natural frequencies of a cracked beam (FNF, SNF, and TNF) are used as inputs of the crack discovery problem to measure the objective function. Also, the weighting factors, \phi_i’s, are considered as {\displaystyle \frac{1}{I}} [52,53]. PSO is an evolutionary optimization method offered by Kennedy and Eberhart [54]. PSO uses a society-dependent universal probe method in which every single particle behaves as particle of the flock to apportion data between them such as to obtain a universal optimum. GA is a heuristically probing algorithm based on natural selection. GA forms an intelligent expansion of a random probe within a determined search area to solve a problem. This algorithm uses a society of persons that are subject to mutability-compelling factors like mutation and crossover. Also, a compatibility function is utilized to appraise persons and reproductive achievement changes with compatibility. FA is based on the behavior of fireflies, a type of insect. Most fireflies produce rhythmical and partial sparkles [55,56]. GWO has been widely applied to many optimization problems due to its advantages over other swarm intelligence techniques. Moreover, the leadership, hierarchy, and quarrying craft of grey wolves in nature are displayed. Also, four kinds of grey wolves denoted as omega (ω), delta (δ), beta (β), and alpha (α) are offered to model the hierarchy [57].

3.1 Hybrid algorithms functions

This section explains the new hybridized algorithms involving PSO, GA, FA, and GWO algorithms. The proposed triple hybrid algorithms (PSO-GA-FA, PSO-GWO-FA, PSO-GA-GWO) have been extended without changing the basic operation of the PSO, FA, GA, and GWO algorithms. It is already known that the PSO algorithm achieves better results in almost all real-world problems; but existence of a solution is needed to reduce the possibility that the PSO algorithm will become trapped in a local minimum. In the proposed method, the GWO, FA, and GA algorithms are utilized to support the PSO algorithm to decrease the likelihood of falling into a local minimum.

3.1.1 Particle Swarm Optimization-Genetic Algorithm-Firefly Algorithm

The intention of combining three evolutionary algorithms PSO, GA, and FA to create the hybrid PSO-GA-FA is based on natural selection. GA that makes the ‘offspring’ seize the favored genetic structure from parents. In this hybrid algorithm, PSO obtains a good outcome for every individual, and these experiences aid FA to get a better survival chance in the symbiotic interplay. As the basic actions in the natural selection repeatedly do, it requires to run the GA, PSO, and FA algorithms consecutively to simulate these actions. Fig.2 presents the schematic view of the proposed PSO-GA-FA algorithm. As can be seen, this method consists of 3 main phases (PSO, GA, and FA), which are executed consecutively.

3.1.2 Particle Swarm Optimization-Genetic Algorithm-Grey Wolf Optimization

The theory of compounding GWO, PSO, and GA are considered for dominating problems of the algorithms mentioned above and getting the accurate response for recognition of a crack in a cantilever beam. PSO-GA-GWO together can provide better results than PSO, GWO, and GA individually. Firstly, determining the objective function, variables, and any parameter of the algorithm is essential. This method searches the optimum answer till the stopping criteria are found, or repetition ends location. When appraisal and repetition of PSO ends, GA begins acting by solving optimization problems involving people selection, crossover, and mutation. Thereafter, GWO starts its action. The sequence of actions of the suggested algorithm PSO-GA-GWO is shown in Fig.3.

3.1.3 Particle Swarm Optimization-Grey Wolf Optimization-Firefly Algorithm

By embedding the GWO and FA operators in PSO, equilibrium among the discovery and extraction capability is amended better. First of all, the objective function, variables, and any algorithm parameter should be determined. This procedure searches the optimum values of objective function by updating the location and velocity of the results till the stopping criteria are found, or repetition ends. When assessment and repetition of the results ends in PSO, GWO begins to act and carry on to solve the optimization problem. Then, FA starts its action by assessing fireflies’ brightness and updating them until the repetition ends. The process of optimizing with the PSO-GWO-FA algorithm is displayed in Fig.4. Also, the pseudocode of the proposed algorithms is described in Pseudocode 1. On this pseudocode, first main subroutine runs and then first PSO subroutine runs and by attention to the algorithms, the following subroutine runs.

4 Discussion

The geometrical and mechanical properties of the considered beam are introduced in Tab.2. The natural frequencies of the Euler-Bernoulli beam for various locations and depths of cracks are presented in Tab.3. In this paper, the method applied in Ref. [58]. is used to determine the natural frequencies of a cracked Euler-Bernoulli beam. In this section, the technique of detecting cracks in the cantilever beam is presented by the proposed algorithms. It aims to detect location and depth of a crack by optimizing the objective function based on the natural frequencies of the beam. Thus, the results of the crack detection in the cracked beams are reported by the presented algorithms. In this paper, five algorithms (GA, GWO, FA, PSO, and MPSO) were compared.

Three sets of the control parameters are chosen for evaluating various strategies of the proposed algorithms, as explained in Tab.4. In each set, the number of populations is different, so sets A, B, and C equal 10, 21, and 32, respectively. The control parameters of the PSO algorithm such as inertia weight parameter, W, Cognitive parameter, C₁, and Social parameter, C₂, have the same values in all sets and other parameters have different values. To compare the achieved results of the present study with the results of Jena and Parhi [36], the properties of the cantilever beam based on their data are considered. The proposed triple hybrid algorithms are investigated by comparison with the results of Jena and Parhi [36] for six cases. These comparisons are shown in Tab.5 and Tab.6 for set A. Tab.7 and Tab.8 show the results of PSO, GA, FA, and GWO algorithms in identification of crack location and depth for set A. To demonstrate the effectiveness of the proposed triple hybrid algorithms, a performance index is defined. This index reports the variance between evaluated and actual values of parameters. This performance index is defined as follows:

(6)\mathrm{\%}error=\left|\frac{D_2-D_1}{D_2}\right|\times 100.

The index is applied to comparing the results from the proposed algorithms. Tab.5 and Tab.6 suggest that the performance of the PSO-GA-FA algorithm is better than the PSO-GA-GWO algorithm and that is better than the PSO-GWO-FA algorithm. The results show that the average calculation errors for crack location and depth equivalent, respectively, are 2.04 × 10⁻³% and 2.79 × 10⁻¹% for the PSO-GA-FA algorithm while reaching 1.1 × 10⁻³% and 2.62 × 10⁻¹% for the PSO-GA-GWO algorithm, 1.81 × 10⁻²% and 1.54 × 10⁻¹% for the PSO-GWO-FA algorithm. Finally, based on findings in this study, the crack location and depth are more accurately determined by the proposed algorithms. Also, Tab.7 and Tab.8 indicate that PSO performs better than GWO, and GWO is higher in convergence GA, with the performance indexes for crack location and depth equivalent to 2.73 × 10⁻⁴% and 2.01 × 10⁻¹% for PSO while reaching 4.97 × 10⁻¹% and 9.73 × 10⁻¹% for GA, 3.24 × 10⁻¹% and 2.94 × 10⁻¹% for GWO, and 3.25 × 10⁻²% and 2.0 × 10⁻¹% for FA.

Fig.5 displays the convergence of algorithms based on set A for the cracked beam model via the existence of crack case no. 6. Fig.5(a) clearly shows that FA has a higher convergence than others. On the other hand, Fig.5(b) illustrates that the PSO-GA-FA algorithm has the best convergence in comparison with the proposed algorithm in the identification of the location and depth of cracks.

Fig.6 shows the convergence characteristics of the proposed triple algorithms that have the most errors for crack detection. In Fig.7-Fig.10, the convergences of the triple hybrid algorithms for the cracked beam model to find the crack location and depth for cases no. 2, 4, 8, and 10 are shown for all sets of control parameters. Also, the numerical results of the identification of the crack location and depth utilizing triple algorithms that are based on collections A to C are shown in Tab.9 and Tab.10 for cases no. 2, 4, 8, and 10. Finally, Tab.11 shows the best algorithm in each set for all cases.

Tab.12 shows the standard deviation (SD) values of the best cost obtained after five independent performances of the PSO-GWO-FA algorithm and the computational time required for each run. It should be noted that all the results obtained and reported in Tables and Figures are based on assumptions of Tab.4. Tab.4 and the obtained results show that the proposed algorithms have better results when population and number of iterations are both larger. Also, population increase is an effective parameter in improving the performance of algorithms; increase in the number of iteration causes better convergency in the proposed algorithms.

5 Conclusions

In this study, novel optimization algorithms are presented for crack detection in a cantilever Euler-Bernoulli beam. Methods using PSO-GWO-FA, PSO-GA-GWO, and PSO-GA-FA algorithms were created by modifying and improving PSO, GA, FA, and GWO algorithms to enhance their accuracy and speed of convergence. This paper explores variations between evaluated frequencies by the proposed algorithm and the measured frequencies for a cracked beam. The results of these proposed hybrid algorithms is compared with results of GA, GWO, FA, PSO, and MPSO. It is shown that the PSO-GA-FA algorithm in set A, the PSO-GWO-FA algorithm in set B, and the PSO-GWO-FA algorithm in set C have good convergency for all cases. Therefore, it is concluded that the PSO-GA-FA and the PSO-GWO-FA algorithms have good accuracy in crack detection. The presented results show that the error in the crack detection using the proposed triple hybrid algorithms is approximately zero and the proposed algorithms provide improved accuracy relative to other algorithms, including those presented in previous studies in the identification of crack location and depth. These triple hybrid algorithms can be used for crack detection in structures under complex loadings.

6 Notations

PSO: Particle Swarm Optimization

GA: Genetic Algorithm

FA: Firefly Algorithm

GWO: Grey Wolf Optimization

EI: Flexural rigidity

PSO-GA-FA: Particle Swarm Optimization-Genetic Algorithm-Firefly Algorithm

PSO-GWO-FA: Particle Swarm Optimization-Grey Wolf Optimization-Firefly Algorithm

PSO-GA-GWO: Particle Swarm Optimization-Genetic Algorithm-Grey Wolf Optimization

7 Appendix

B-Matrix is defined as:

\left|\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 1& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -S_1& S_2\\ 0& 0& 0& 0& S_3& S_4& -S_5& -S_6& 0& 0& 0& 0\\ 0& 0& 0& 0& S_4& S_3& S_6& -S_5& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& S_7& S_8& -S_7& -S_8\\ S_9& S_{10}& S_{11}& S_{12}& -S_9& -S_{10}& -S_{11}& -S_{12}& 0& 0& 0& 0\\ S_9& S_{10}& -S_{11}& -S_{12}& -S_9& -S_{10}& S_{11}& S_{12}& 0& 0& 0& 0\\ S_{10}& S_9& S_{12}& -S_{11}& -S_{10}& -S_9& -S_{12}& S_{11}& 0& 0& 0& 0\\ S_{13}& S_{14}& S_{15}& S_{16}& S_{17}& S_{18}& S_{19}& S_{20}& S_{21}& S_{22}& S_{23}& S_{24}\\ S_{25}& S_{26}& S_{27}& S_{28}& S_{29}& S_{30}& S_{31}& S_{32}& S_{33}& S_{34}& S_{35}& S_{36}\end{array}\right|

where

S_1 = sin H_{\mathrm{s}}, S_2 = cos H_{\mathrm{s}}, S_3 = cosh H_{\mathrm{v}},

S_4 = sinh H_{\mathrm{v}}, S_5 = cos H_{\mathrm{v}}, S_6 = sin H_{\mathrm{v}},

S_7 = cos (a\times H_{\mathrm{s}}), S_8 = sin (a\times H_{\mathrm{s}}), S_9 = cosh (a\times H_{\mathrm{v}}),

S_{10} = sinh (a\times H_{\mathrm{v}}), S_{11} = cos (a\times H_{\mathrm{v}}), S_{12} = sin (a\times H_{\mathrm{v}}),

M_1 = H_{\mathrm{v}}\times P₁, M_2 = H_{\mathrm{s}}\times P_2, M_3 = H_{\mathrm{v}}\times P_4,

M_4 = H_{\mathrm{s}}\times P₁, M_5 = H_{\mathrm{v}}\times P₃, S_{13} = M_1\times S_{10},

S_{14} = M_1\times S_9, S_{15} = − M_1\times S_{12}, S_{16} = M_1\times S_{11},

S_{17} = − M_1\times S_{10}, S_{18} = − M_1\times S_9, S_{19} = M_1\times S_{12},

S_{20} = − M_1\times S_{11}, S_{21} = (− M_2\times S_8 – M_2\times P_1 \times S_8), S_{22} = (M_2\times S_7+ M_4\times P_2\times S_7),

S_{23} = M_2\times S_8, S_{24} = − M_2\times S_7, S_{25} = (M_3\times S_{10}+M_5^2\times P_4\times S_9),

S_{26} = (M_3\times S_9 + M_3^2\times P_3\times S_{10}), S_{27} = (− M_3\times S_{12} − M_5^2\times P_4\times S_{11}), S_{28} = (M_3\times S_{11}-M_5^2\times P_3 \times S_{12}),

S_{29} = − M_3\times S_{10}, S_{30} = − M_3\times S_9, S_{31} = M_3\times S_{12},

S_{32} = − M_3\times S_{11}, S_{33} = P_3\times S_7, S_{34} = P_3\times S_8,

S_{35} = − P_3\times S_7, S_{36} = − P_3\times S_8, P_1 = {\displaystyle \frac{AE}{L{K}_{11}}},

P_2 = {\displaystyle \frac{AE}{K_{12}}}, P_3 = {\displaystyle \frac{EI}{L{K}_{22}}}, P_4 = {\displaystyle \frac{EI}{L^2{K}_{21}}}.

References

[1] Dimarogonas A, Papadopoulos C. Vibration of cracked shafts in bending. Journal of Sound and Vibration, 1983, 91(4): 583–593

[2] Qian G L, Gu S N, Jiang J S. The dynamic behaviour and crack detection of a beam with a crack. Journal of Sound and Vibration, 1990, 138(2): 233–243

[3] Nahvi H, Jabbari M. Crack detection in beams using experimental modal data and finite element model. International Journal of Mechanical Sciences, 2005, 47(10): 1477–1497

[4] Chondros T, Dimarogonas A, Yao J. A continuous cracked beam vibration theory. Journal of Sound and Vibration, 1998, 215(1): 17–34

[5] Kim J T, Stubbs N. Crack detection in beam-type structures using frequency data. Journal of Sound and Vibration, 2003, 259(1): 145–160

[6] Orhan S. Analysis of free and forced vibration of a cracked cantilever beam. NDT & E International, 2007, 40(6): 443–450

[7] Saavedra P, Cuitino L. Crack detection and vibration behavior of cracked beams. Computers & Structures, 2001, 79(16): 1451–1459

[8] Zheng D Y, Kessissoglou N. Free vibration analysis of a cracked beam by finite element method. Journal of Sound and Vibration, 2004, 273(3): 457–475

[9] Rizos P, Aspragathos N, Dimarogonas A. Identification of crack location and magnitude in a cantilever beam from the vibration modes. Journal of Sound and Vibration, 1990, 138(3): 381–388

[10] Sahoo B, Maity D. Damage assessment of structures using hybrid neuro-genetic algorithm. Applied Soft Computing, 2007, 7(1): 89–104

[11] Vakil Baghmisheh M T, Peimani M, Sadeghi M H, Ettefagh M M, Tabrizi A F. A hybrid particle swarm–Nelder–Mead optimization method for crack detection in cantilever beams. Applied Soft Computing, 2012, 12(8): 2217–2226

[12] Vakil-Baghmisheh M T, Peimani M, Sadeghi M H, Ettefagh M M. Crack detection in beam-like structures using genetic algorithms. Applied Soft Computing, 2008, 8(2): 1150–1160

[13] Patil D, Maiti S. Experimental verification of a method of detection of multiple cracks in beams based on frequency measurements. Journal of Sound and Vibration, 2005, 281(1−2): 439–451

[14] Rosales M B, Filipich C P, Buezas F S. Crack detection in beam-like structures. Engineering Structures, 2009, 31(10): 2257–2264

[15] Moezi S A, Zakeri E, Zare A, Nedaei M. On the application of modified cuckoo optimization algorithm to the crack detection problem of cantilever Euler–Bernoulli beam. Computers & Structures, 2015, 157: 42–50

[16] Nandwana B, Maiti S. Detection of the location and size of a crack in stepped cantilever beams based on measurements of natural frequencies. Journal of Sound and Vibration, 1997, 203(3): 435–446

[17] Lele S, Maiti S. Modelling of transverse vibration of short beams for crack detection and measurement of crack extension. Journal of Sound and Vibration, 2002, 257(3): 559–583

[18] Viola E, Federici L, Nobile L. Detection of crack location using cracked beam element method for structural analysis. Theoretical and Applied Fracture Mechanics, 2001, 36(1): 23–35

[19] Rezanezhad M, Lajevardi S A, Karimpouli S. An investigation on prevalent strategies for XFEM-based numerical modeling of crack growth in porous media. Frontiers of Structural and Civil Engineering, 2021, 15(4): 914–936

[20] Rungamornrat J, Chansavang B, Phongtinnaboot W, Van C N. Investigation of Generalized SIFs of cracks in 3D piezoelectric media under various crack-face conditions. Frontiers of Structural and Civil Engineering, 2020, 14(2): 280–298

[21] Zainud-Deen S H, Hassen W M, Awadalla K H. Crack detection using a hybrid finite difference frequency domain and particle swarm optimization techniques. In: 2009 National Radio Science Conference. Cairo: IEEE, 2009, 1–8

[22] Nanthakumar S S, Lahmer T, Rabczuk T. Detection of flaws in piezoelectric structures using extended FEM. International Journal for Numerical Methods in Engineering, 2013, 96(6): 373–389

[23] Nanthakumar S S, Lahmer T, Zhuang X, Zi G, Rabczuk T. Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering, 2016, 24(1): 153–176

[24] Nanthakumar S S, Lahmer T, Rabczuk T. Detection of multiple flaws in piezoelectric structures using XFEM and level sets. Computer Methods in Applied Mechanics and Engineering, 2014, 275: 98–112

[25] Samanta S, Nanthakumar S S, Annabattula R K, Zhuang X. Detection of void and metallic inclusion in 2D piezoelectric cantilever beam using impedance measurements. Frontiers of Structural and Civil Engineering, 2019, 13(3): 542–556

[26] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[27] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A geometrically non-linear three-dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75(16): 4740–4758

[28] Ghasemi H, Park H S, Rabczuk T. A level-set based IGA formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 239–258

[29] Ghasemi H, Park H S, Rabczuk T. A multi-material level set-based topology optimization of flexoelectric composites. Computer Methods in Applied Mechanics and Engineering, 2018, 332: 47–62

[30] Ghasemi H, Kerfriden P, Bordas S P A, Muthu J, Zi G, Rabczuk T. Interfacial shear stress optimization in sandwich beams with polymeric core using non-uniform distribution of reinforcing ingredients. Composite Structures, 2015, 120: 221–230

[31] Ghasemi H, Kerfriden P, Bordas S P A, Muthu J, Zi G, Rabczuk T. Probabilistic multiconstraints optimization of cooling channels in ceramic matrix composites. Composites. Part B, Engineering, 2015, 81: 107–119

[32] Talebi H, Silani M, Rabczuk T. Concurrent multiscale modeling of three dimensional crack and dislocation propagation. Advances in Engineering Software, 2015, 80: 82–92

[33] Zhou S, Rabczuk T, Zhuang X. Phase field modeling of quasi-static and dynamic crack propagation: COMSOL implementation and case studies. Advances in Engineering Software, 2018, 122: 31–49

[34] Zhou S, Zhuang X, Zhu H, Rabczuk T. Phase field modelling of crack propagation, branching and coalescence in rocks. Theoretical and Applied Fracture Mechanics, 2018, 96: 174–192

[35] Zhou S, Zhuang X, Rabczuk T. A phase-field modeling approach of fracture propagation in poroelastic media. Engineering Geology, 2018, 240: 189–203

[36] Jena P K, Parhi D R. A modified particle swarm optimization technique for crack detection in Cantilever Beams. Arabian Journal for Science and Engineering, 2015, 40(11): 3263–3272

[37] Ghannadiasl A, Ajirlou S K. Analytical solution of dynamic analysis of cracked Euler–Bernoulli beam with elastic boundary condition by GFM. Romanian Journal of Acoustics and Vibration, 2018, 15(2): 100–107

[38] Ghadimi S, Kourehli S S. Crack detection of structures using modified extreme learning machine (MELM). Inverse Problems in Science and Engineering, 2017, 25(7): 995–1013

[39] Ghadimi S, Kourehli S S. Multiple crack identification in Euler beams using extreme learning machine. KSCE Journal of Civil Engineering, 2017, 21(1): 389–396

[40] Prawin J, Rama Mohan Rao A. Reference-free breathing crack identification of beam-like structures using an enhanced spatial Fourier power spectrum with exponential weighting functions. International Journal of Structural Stability and Dynamics, 2019, 19(2): 1950017

[41] Ghadimi S, Kourehli S S. Multi cracks detection in Euler-Bernoulli beam subjected to a moving mass based on acceleration responses. Inverse Problems in Science and Engineering, 2018, 26(12): 1728–1748

[42] Samir K, Brahim B, Capozucca R, Abdel Wahab M. Damage detection in CFRP composite beams based on vibration analysis using proper orthogonal decomposition method with radial basis functions and cuckoo search algorithm. Composite Structures, 2018, 187: 344–353

[43] Khatir S, Dekemele K, Loccufier M, Khatir T, Abdel Wahab M. Crack identification method in beam-like structures using changes in experimentally measured frequencies and Particle Swarm Optimization. Comptes Rendus Mécanique, 2018, 346(2): 110–120

[44] Wimarshana B, Wu N, Wu C. Application of entropy in identification of breathing cracks in a beam structure: Simulations and experimental studies. Structural Health Monitoring, 2018, 17(3): 549–564

[45] Wei Z, Liu J, Lu Z. Structural damage detection using improved particle swarm optimization. Inverse Problems in Science and Engineering, 2018, 26(6): 792–810

[46] Khatir S, Abdel Wahab M, Boutchicha D, Khatir T. Structural health monitoring using modal strain energy damage indicator coupled with teaching-learning-based optimization algorithm and isogoemetric analysis. Journal of Sound and Vibration, 2019, 448: 230–246

[47] Zenzen R, Belaidi I, Khatir S, Abdel Wahab M. A damage identification technique for beam-like and truss structures based on FRF and Bat Algorithm. Comptes Rendus Mécanique, 2018, 346(12): 1253–1266

[48] Wang T, Noori M, Altabey W A. Identification of cracks in an Euler–Bernoulli beam using Bayesian inference and closed-form solution of vibration modes. Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 2021, 235(2): 421–438

[49] Chinka S S B, Putti S R, Adavi B K. Modal testing and evaluation of cracks ofdn cantilever beam using mode shape curvatures and natural frequencies. Structures, 2021, 32(1): 1386–1397

[50] Wu Z, Huang B, Tee K F, Zhang W. A novel stochastic approach for static damage identification of beam structures using homotopy analysis algorithm. Sensors (Basel), 2021, 21(7): 2366

[51] Broumand P. Inverse problem techniques for multiple crack detection in 2D elastic continua based on extended finite element concepts. Inverse Problems in Science and Engineering, 2021, 29(12): 1702–1728

[52] Casciati S. Stiffness identification and damage localization via differential evolution algorithms. Structural Control and Health Monitoring, 2008, 15(3): 436–449

[53] Casciati S, Elia L. Potential of two metaheuristic optimization tools for damage localization in civil structures. Journal of Aerospace Engineering, 2017, 30(2): B4016012

[54] Kennedy J, Eberhart R. Particle swarm optimization. In: Proceedings of ICNN'95—International Conference on Neural Networks. Perth: IEEE, 1995, 1942–1948

[55] Yang X S. Firefly algorithm, stochastic test functions and design optimisation. International Journal of Bio-inspired Computation, 2010, 2(2): 78–84

[56] Yang X S. Firefly algorithms for multimodal optimization. In: International Symposium on Stochastic Algorithms. Berlin: Spriner, 2009, 169–178

[57] Mirjalili S, Mirjalili S M, Lewis A. Grey wolf optimizer. Advances in Engineering Software, 2014, 69: 46–61

[58] Ghannadiasl A, Khodapanah Ajirlou S. Forced vibration of multi-span cracked Euler–Bernoulli beams using dynamic Green function formulation. Applied Acoustics, 2019, 148: 484–494

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